minimal polynomial and relevance

Question

Suppose that we had $K[\alpha]$ and $K[\beta]$ as simple extensions over $K$. I have read that $[K(\alpha,\beta), K(\alpha)] \leq [K(\beta):K]$ and the reason for this was because the minimal polynomial divides something. I'm not sure what that something is and/or how the minimal polynomial is coming into play.

Answer 1

By definition, $[K(\beta):K]$ is the degree of the minimal polynomial $f_\beta$ of $\beta$ over $K$ and $[K(\alpha,\beta):K(\alpha)]$ is the degree of the minimal polynomial $g_\beta$ of $\beta$ over $K(\alpha)$.

Since $f_\beta$ also belongs to $K(\alpha)[X]$ and $f_\beta(\beta)=0$, we have by definition that $g_\beta \mid f_\beta$. Thus $deg(g_\beta)<deg(f_\beta)$.